Linear Interpolation: Finding Values Between Table Entries
October 29, 2026
Tables list values at specific points, but real inputs often fall between the entries: a pipe pressure between two tabulated temperatures, a tax bracket between published income points, or a tire pressure for a load between two chart rows. Linear interpolation estimates the value in between by assuming the quantity changes in a straight line between the two nearest entries.
The formula is y = y₀ + (x − x₀) × (y₁ − y₀) ÷ (x₁ − x₀), where (x₀, y₀) and (x₁, y₁) are the neighboring table entries. If a table gives 100 at x = 30 and 120 at x = 40, then at x = 37, y = 100 + (37 − 30) × (120 − 100) ÷ (40 − 30) = 100 + 7 × 2 = 114.
Skip the math and get your answer instantly:
Open the Scientific Calculator →Worked Example: Fuel Consumption Chart
A chart lists fuel use of an engine as 5.5 gallons per hour at 2 thousand rpm and 7.0 at 3 thousand rpm. At 2,700 rpm, the fraction of the way from 2,000 to 3,000 is (2.7 − 2) ÷ (3 − 2) = 0.7, so fuel use = 5.5 + 0.7 × (7.0 − 5.5) = 5.5 + 1.05 = 6.55 gallons per hour. In other words, find how far you are between the two x values (as a fraction), and move the same fraction between the two y values.
The same calculation works in a spreadsheet using the FORECAST function or a formula, and interpolation is built into lookup tools.
| Given points | Target x | Fraction along | Interpolated y |
|---|---|---|---|
| (30, 100) and (40, 120) | 37 | 0.7 | 114 |
| (2, 5.5) and (3, 7.0) | 2.7 | 0.7 | 6.55 |
| (10, 4.0) and (20, 3.0) | 12 | 0.2 | 3.8 |
| (0, 32) and (100, 212) | 37 | 0.37 | 98.6 |
Accuracy: When Straight Lines Fit
Linear interpolation is exact when the true relationship is a straight line, as with temperature conversion (the last row gives 98.6°F for 37°C). For curved relationships, it is accurate only if the table entries are close together. For a function like y = x² between x = 2 (y = 4) and x = 4 (y = 16), interpolating at x = 3 gives 10, while the true value is 9, an error of 11%. Halving the spacing to x = 2.5 and 3.5 (6.25 and 12.25) gives 9.25 at x = 3, an error of 3%: the error shrinks with the square of the spacing. For strongly curved data, use a finer table, logarithmic interpolation for exponentially changing quantities, or fitted curves.
Extrapolation Is Riskier
Extrapolation extends the line beyond the table's range. It uses the same formula but is much less reliable because nothing constrains the real curve outside the known points. A table of 10-year growth that ends at year 10 is a poor basis for a forecast at year 30. Clearly label extrapolated values, keep them close to the data, and check them against other information.
Practical Uses
Tax and rate schedules that list values at selected income levels; engineering handbooks with steam tables and pipe friction charts; the lookup of unit conversions between points; calibration curves in laboratories; computer graphics (blending colors and positions); and filling in missing time-series data (a sensor reading at 10:00 and 10:20 gives an estimate at 10:05). In finance, interpolating between known yields on 5-year and 10-year bonds gives an estimate for 7 years: if the yields are 3.8% and 4.3%, the 7-year yield is 3.8 + 0.4 × 0.5 = 4.0%.
Frequently Asked Questions
What is the linear interpolation formula?
y = y₀ + (x − x₀)(y₁ − y₀)/(x₁ − x₀).
When is linear interpolation accurate?
When the underlying relationship is nearly linear between the points, or the points are close together.
What is the difference between interpolation and extrapolation?
Interpolation estimates inside the data range; extrapolation estimates beyond it and is less reliable.
How can I interpolate in a spreadsheet?
Use the FORECAST function with the two nearest points, or write the formula directly.
What if the data are curved?
Use more closely spaced points or a curve-fitting method.